Trigonometry

Maths Pure 3 · Trigonometry. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.

Test this topic free All Maths Pure 3 topics

The idea

The new functions sec, cosec and cot are the reciprocals of cos, sin and tan. Their graphs have vertical asymptotes wherever the original function is zero. Because cos θ lies between −1 and 1, sec θ is never between −1 and 1, and the same is true for cosec θ.

The rest of the topic is a set of identities: two Pythagorean ones for sec and cosec, the compound angle formulae for sin(A ± B), cos(A ± B) and tan(A ± B), and the double angle formulae that follow from them.

The expression a sin θ + b cos θ can be written as one single wave, R sin(θ + α). This makes it easy to find its greatest value or to solve an equation.

Rules to remember

sec θ = 1/cos θ, cosec θ = 1/sin θ, cot θ = 1/tan θ = cos θ/sin θ
sec²θ = 1 + tan²θ and cosec²θ = 1 + cot²θ
sin(A ± B) = sin A cos B ± cos A sin B
cos(A ± B) = cos A cos B ∓ sin A sin B (the sign in the middle is the opposite one)
tan(A ± B) = (tan A ± tan B)/(1 ∓ tan A tan B)
sin 2A = 2 sin A cos A; cos 2A = cos²A − sin²A = 2cos²A − 1 = 1 − 2sin²A; tan 2A = 2 tan A/(1 − tan²A)
For a, b > 0: a sin θ + b cos θ = R sin(θ + α), where R = √(a² + b²) and tan α = b/a; the greatest value is R and the least is −R

Common mistake

In cos(A + B) students keep the plus sign and write cos A cos B + sin A sin B. For cosine the sign changes: cos(A + B) = cos A cos B − sin A sin B. Test with A = B = 45°: the left side is cos 90° = 0.

Worked example

Express 5 sin θ + 12 cos θ in the form R sin(θ + α), where R > 0 and 0° < α < 90°, and state its greatest value.

R sin(θ + α) = R cos α sin θ + R sin α cos θ, so R cos α = 5 and R sin α = 12.
R = √(5² + 12²) = √169 = 13.
tan α = 12/5 = 2.4, so α = 67.4°.
The greatest value of sin(θ + α) is 1, so the greatest value is 13 × 1 = 13.

Answer: 13 sin(θ + 67.4°); greatest value 13

Ask about this

Type what you are stuck on. The helper finds the explanation, worked examples and the notes to read, free and instant.

Read the full notes

Chapter 3: Trigonometry

Keep going

← Logarithmic and exponential functions Differentiation →Take a Maths Pure 3 test